This paper focuses on the following class of fractional magnetic Schr\"{ o}dinger equations \begin{ equation*} (-\Delta)_{ A}^{ s}u+V(x)u=g(\vert u\vert^{ 2})u+\lambda\vert u\vert^{ q-2}u, \quad \mbox{ in } \mathbb{ R}^{ N}, \end{ equation*} where $(-\Delta)_{ A}^{ s}$ is the fractional magnetic Laplacian, $A :\mathbb{ R}^N \rightarrow \mathbb{ R}^N$ is the magnetic potential, $s\in (0,1)$, $N>2s$, $\lambda \geq0$ is a parameter, $V:\mathbb{ R}^N \rightarrow \mathbb{ R}$ is a potential function that may decay to zero at infinity and $g: \mathbb{ R}_{ +} \rightarrow \mathbb{ R}$ is a continuous function with subcritical growth. We deal with supercritical case $q\geq 2^*_s:=2N/(N-2s)$. Our approach is based on variational methods combined with penalization technique and $L^{ \infty}$-estimates.
Categories: math.AP